308 Appendix
and letInandJndenote arrangements or permutations of the samen
integers
In={
i 1 i 2 i 3 ···in}
,
Jn={
j 1 j 2 j 3 ···jn}
.
There aren! possible sets of the formInorJnincludingNn. The num-
bers within the set are called elements. The operation which consists of
interchanging any two elements in a set is called an inversion. Assuming
thatJn=In, that is,jr=irfor at least two values ofr, it is possible to
transformJnintoInby means of a sequence of inversions. For example, it
is possible to transform the set{ 35214 }into the setN 5 in four steps,
that is, by means of four inversions, as follows:
35214
1: 15234
2: 12534
3: 12354
4: 12345
The choice of inversions is clearly not unique for the transformation can
also be accomplished as follows:
35214
1: 34215
2: 31245
3: 21345
4: 12345
No steps have been wasted in either method, that is, the methods are
efficient and several other efficient methods can be found. If steps are wasted
by, for example, removing an element from its final position at any stage
of the transformation, then the number of inversions required to complete
the transformation is increased.
However, it is known that if the number of inversions required to trans-formJnintoInis odd by one method, then it is odd by all methods, and
Jnis said to be an odd permutation ofIn. Similarly, if the number of in-
versions required to transformJnintoInis even by one method, then it is
even by all methods, andJnis said to be an even permutation ofIn.
The permutation symbol is an expression of the form{
InJn}
=
{
i 1 i 2 i 3 ··· inj 1 j 2 j 3 ··· jn}
,
which enablesInto be compared withJn.
The sign of the permutation symbol, denoted byσ, is defined as follows:σ= sgn{
InJn}
= sgn{
i 1 i 2 i 3 ··· inj 1 j 2 j 3 ··· jn}
=(−1)
m
,